In my Proton organization, a managed member (an account I created, not admin, not private) cannot set up the Gmail connection. In Settings > Import via Easy Switch > Google, the flow always creates "Google Message (forwarding only)". Deleting it and starting again gives the same result. The member receives Gmail mail in Proton but cannot send from the Gmail address. On a standalone Proton account, the same steps create the full connection.

The web client blocks this on purpose: a managed user gets the connection only as an admin.

https://github.com/ProtonMail/WebClients/blob/bc6f23c7a8e1ee29319e44d9210deca9a5d49781/packages/activation/src/helpers/byoeAccess.ts#L6-L20

https://github.com/ProtonMail/WebClients/blob/bc6f23c7a8e1ee29319e44d9210deca9a5d49781/packages/activation/src/helpers/byoeAccess.test.ts#L51-L55

The only workaround is to make the member an organization admin. That needs the organization password and gives admin rights over the whole organization.

Proton's support page says: "Gmail forwarding is a legacy feature that is gradually being replaced by Gmail connection". Managed members stay on that legacy feature.

https://proton.me/support/automatic-forwarding-gmail

Please let managed members set up the Gmail connection, or let an admin allow it per member in Users and addresses > Edit user.